Olympiad Number Theory · Superchallenge problem

Triangle $ABC$ has $A = (0,0)$, $B = (4,0)$, $C = (0,3)$, and $P = (4,3)$ lies on its circumcircle. The feet of the perpendiculars from $P$ to lines $AB$, $BC$, $CA$ are collinear (the Simson line of $P$). Compute the distance between the feet on $AB$ and on $CA$.